CSE20

Lecture 2: Number Systems: Binary Numbers and Gray Code

CK Cheng

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Number Systems

1. Introduction2. Binary Numbers3. Gray code4. Negative Numbers5. Residual Numbers

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2. Binary Numbers: iClickerWhat is the extent of a binary number system• A. Coverage of integer and floating point

numbers• B. Mechanism of addition and subtraction

operations• C. Operations of logic functions • D. All of the above• E. None of the above.

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2. Binary Numbers

1. Definition (radix 2)2. Enumerations (value -> index)3. Addition (logic -> hardware)

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2.1 Definition of Binary Numbers

• Format: An n digit binary number (bn-1, …, b1, b0)2 where bi in {0,1} for 0<= i < n

• Value: bn-12n-1+ …+b121+b020

• Non-redundancy: The system is non-redundant, i.e. different binary numbers represent different values.

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2.2 Enumeration of Binary Numbers

id b2b1b0

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

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id b1b0

0 00

1 01

2 10

3 11

id b0

0 0

1 1

1 digit 2 digits 3 digits 4 digits?

id b3b2b1b0

0 0000

1 0001

2 0010

3 0011

. .

. .

. .

14 1110

15 1111

An n digit binary code covers numbers from0 to 2n-1.

2.2 Enumeration of binary numbers iCliker

When we enumerate binary numbers(b3b2b1b0)2 from 0 to 15, the sequence of b3

should be• A. 0101010101010101• B. 0011001100110011• C. 0000111100001111• D. 0000000011111111

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2.3 Addition of Binary Numbers

Given two binary numbers A & B, we derive binary number S so that the value of S is equal to the sum of the values of A & B, i.e.

(an-1…,a1a0)2+(bn-1…b1b0)2=(sn-1…s1s0)2

Caution: Overflow, i.e. the sum is beyond the range of the representation.

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2.3 Addition: iClickerGiven two binary numbers A=(an-1…,a1a0)2 and B=(bn-1…b1b0)2

what is the largest possible value of A+B?A.2n+1

B.2n+1-1C.2n+1-2D.None of the above

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2.3 Addition of Binary NumbersEquality of addition (an-1…,a1a0)2+(bn-1…b1b0)2=(sn-1…s1s0)2

That is to sayan-12n-1+…+a121+a020+bn-12n-1+…+b121+b020

=(an-1+bn-1)2n-1+…+(a1+b1)21+(a0+b0)20

=sn-12n-1+…+s121+s020

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2.3 Addition of Binary Numbers

b2 b1 b0 Value

0 0 0 0

0 0 1 1

0 1 0 2

0 1 1 3

1 0 0 4

1 0 1 5

1 1 0 6

1 1 1 7

8 4 2 10 0 1 10 1 0 1

8 4 2 10 0 1 10 1 1 0

+

+

Examples:

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2.3 Addition of Binary NumbersBiti+1 Biti Biti-1

Carryi+1 Carryi

ai ai-1

bi bi-1

Sumi Sumi-1

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Formula for Bit i: Carryi+ai+bi= 2xCarryi+1+Sumi

2.3 Adding 2 bits in a digit

a b Carry Sum

0 0 0 00 1 0 11 0 0 11 1 1 0

Formula:

a+b=

2xCarry + Sum

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2.3 Adding 3 bits in a digitid a b c Carry Sum

0 0 0 0 0 01 0 0 1 0 12 0 1 0 0 13 0 1 1 1 04 1 0 0 0 15 1 0 1 1 06 1 1 0 1 07 1 1 1 1 1

Formula:

a+b+c=

2xCarry + Sum

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3. Gray Code

1. Introduction2. Example3. Construction4. Comments

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3.1 Gray Code: IntroductionGray: Frank Gray patented the code in 1947

A variation of binary code

The code will be used for logic operation (CSE20, CSE140)

Feature: only one bit changes for two consecutive numbers

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3.2 Gray Code: Example

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id b2b1b0 g2g1g0

0 000 000

1 001 001

2 010 011

3 011 010

4 100 110

5 101 111

6 110 101

7 111 100

id b1b0 g1g0

0 00 00

1 01 01

2 10 11

3 11 10

2 digits 3 digits

Note the difference of the first and last rows.

3.2 Gray Code

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id b2b1b0 g2g1g0

0 000 000

1 001 001

2 010 011

3 011 010

4 100 110

5 101 111

6 110 101

7 111 100

3 digits

id b3b2b1b0 g3g2g1g0

0 0000 0000

1 0001 0001

2 0010 0011

3 0011 0010

4 0100 0110

5 0101 0111

6 0110 0101

7 0111 0100

8 1000 ?

9 1001

10 1010

11 1011

12 1100

13 1101

14 1110

15 1111

3.2 Gray Code: iClicker

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A 4-digit Gray code (g3g2g1g0) at id=8 is written as (ref: previous page)A.(0101)B.(0110)C.(1100) D.None of the above

3.3 Gray Code: Construction

Construction of n-digit Gray code from n-1 digit Gray code

•Copy the n-1 digit Gray code for the top 2n-1 rows. Fill 0 at digit gn-1 in the top rows.

•Reflect and append the n-1 digit code for the bottom 2n-1 rows. Fill 1 at digit gn-1 in the bottom rows.

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3.4 Gray Code: Comments•There are various codes that satisfy the Gray code feature.

•Gray code saves communication power when the signals are continuous in nature, e.g. addresses, analog signals

•Gray code facilitates code checking when the signals are supposed to be continuous in value.•For arithmetic operations, we need to convert the values.

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